Detecting Gaussian entanglement via extractable work

We show how the presence of entanglement in a bipartite Gaussian state can be detected by the amount of work extracted by a continuous-variable Szilard-like device, where the bipartite state serves as the working medium of the engine. We provide an expression for the work extracted in such a process and specialize it to the case of Gaussian states. The extractable work provides a sufficient condition to witness entanglement in generic two-mode states, becoming also necessary for squeezed thermal states. We extend the protocol to tripartite Gaussian states and show that the full structure of inseparability classes cannot be discriminated based on the extractable work. This suggests that bipartite entanglement is the fundamental resource underpinning work extraction.

Figure 1

(a) Gaussian demons Alice (orange) and Bob (purple) share a bipartite Gaussian state of modes $\widehat{a}$ and $\widehat{b}$ and want to know whether the state is entangled (yellow line) or separable (gray line). In order to do so, they check how much work Alice can extract from a heat bath when only local Gaussian measurements are allowed. In the first strategy (b) Bob performs a Gaussian measurement ${\widehat{\pi }}_b$ and Alice extracts mechanical work by letting her conditional state ${\sigma }_a^{{\pi }_b}$ expand (from orange to red), e.g., pushing the demon's board. As a result of the protocol Alice extracts an amount of work $W$. In the second approach (c), both demons perform a measurement and the work is extracted from the classical register of the results.

Figure 2

Extractable work $W$ (in units of $k_{B}T$) against $a$ for randomly generated states. Each point corresponds to a state obtained by a uniform sampling of the parameters $a$ and $c$. Points corresponding to entangled (separable) states are marked in yellow (gray). (a) Homodyne detection and (b) heterodyne detection. The red dashed curve represents the maximum amount of extractable work $W_{\text{max}}$, while the black solid curve stands for the work at the separability threshold $W_{\text{sep}}^{\left(k\right)}$, $k=0,1$. (c) Extractable work against the parameter $c$ for different Gaussian measurements and $a=3$. Solid, dashed, and dot-dashed curves refer to $\lambda =0$, 5, and 1, respectively. The vertical dashed line refers to the value $c_{\text{sep}}=a-1/2$, while the horizontal ones refer to the corresponding values of $W_{\text{sep}}^{\left(k\right)}$, $k=0,1$.

Figure 3

Extractable work $W$ (in units of $k_{B}T$) against local energies $a$ and $b$ for randomly generated STSs. Each point corresponds to a state obtained by a uniform sampling of the parameters $a$, $b$ and $c$. Points corresponding to entangled (separable) states are marked in yellow (gray). (a) Homodyne detection ($\lambda =0$) and (b) heterodyne detection ($\lambda =1$). Maximum work $W_{\text{max}}^{\left(k\right)}$ (red) and separable work $W_{\text{sep}}^{\left(k\right)}$ (black), $k=0,1$ correspond to red and gray surfaces, respectively. In the right column, sections of both plots are shown.

Figure 4

Extractable work $W$ (in units of $k_{B}T$) for a STS against the parameter $a$. Random generated states are constrained to have $b\le b_{\text{max}}$ where we set $b_{\text{max}}=3$. Points corresponding to entangled (separable) states are marked in yellow (gray) and we performed homodyne detection. The black solid curve is given by $W_{\text{sep}}^{\left(0\right)}$ evaluated at $b=b_{\text{max}}$, while the red dashed one is given by $W_{\text{max}}^{\left(0\right)}$ evaluated at $b=a$.

Figure 5

Extractable work ${\overline{W}}^{\left(\lambda \right)}$ (in units of $k_{B}T$) averaged over the detection angle $\varphi$ against the local energies $a$ and $b$. Points are obtained by random sampling. Detection strength has been fixed to a generic value $\lambda =3$. Points corresponding to entangled (separable) states are marked in yellow (gray). Maximum and separable work $W_{\text{max}}^{\left(\lambda \right)}$ and ${\overline{W}}_{\text{sep}}^{\left(\lambda \right)}$ correspond to red dashed and black solid curves, respectively.

Figure 6

Extractable work $W$ (in units of $k_{B}T$) for symmetric STS against the parameter $c$ and for fixed $a=3$. The red dashed curve is for $W^{(1,1)}$, while the black solid one is for ${\overline{W}}^{(0,0)}$. We also show a comparison with work extracted via single heterodyne detection $W^{\left(1\right)}$ (light red thin dashed curve) and homodyne detection $W^{\left(0\right)}$ (gray thin curve). The vertical dashed line refers to the value $c_{\text{sep}}=a-1/2$.

Figure 7

Extractable work $W$ (in units of $k_{B}T$) against local energy $a$ for a fully symmetric mixed state ${\sigma }_{abc}^S$ when either heterodyne detection (red dot-dashed curve) or homodyne detection (black solid curve) is performed on both Bob's and Charlie's sides. In the case of homodyne detection the work has been averaged over the two angular variable. The black dashed line corresponds to the work extracted from a pure symmetric tripartite state ${\sigma }_{abc}^P$.

Figure 8

Extractable work $W$ (in units of $k_{B}T$) against local energy $a$ for randomly generated tripartite states ${\sigma }_{abc}^M$. Points corresponding to fully inseparable states are marked in yellow, 1-biseparable in red, 2-biseparable in purple, and 3-biseparable in gray. (a)–(c) Homodyne detection ($\lambda =0$) on Bob's and Charlie's sides and (d)–(f) heterodyne detection ($\lambda =1$). The maximum work $W_{\text{max}}$ corresponds to the black curve.